MTH-21b.4 Control Theory for PDE
Syllabus
Module 1: Controllability of ODEs.
- Various concepts of controllability. Reachable states.
- Kalman criterion, Fattorini-Hautus test.
- Various concepts of observability. Duality between controllability and observability.
- Observability inequalities and cost of the control.
- Controllability of non-autonomous linear systems.
- Local and Global Controllability for nonlinear systems in finite dimension.
- Return method.
Module 2: Wellposedness of linear evolution equations via Semigroup theory.
- Time dependent Sobolev spaces, , .
- Unbounded operators and their adjoint.
- Strongly continuous semigroups and generators.
- Theorems of Lumer-Philips, Stone, and Hillie-Yoshida.
- Nonhomogeneous linear evolution equations, Mild Solution.
- Application to transport, heat, and wave equations.
Module 3: Controllability of Linear PDEs.
- Abstract linear control system, admissibility, and wellposedness.
- Controllability and observability concepts. Duality between controllability and observability.
- Controllability of transport equation.
- Fourier Series Methods in control of PDEs, Controllability of 1D heat equation using the method of moments, Ingham inequalities and its applications to wave equations.
- Controllability of wave and heat equations via multiplier method. Carleman estimates.